Optimal. Leaf size=120 \[ -\frac {2^{\frac {3}{2}-\frac {i n}{2}} (1-i a x)^{\frac {1}{2} (3+i n)} \sqrt {c+a^2 c x^2} \, _2F_1\left (\frac {1}{2} (-1+i n),\frac {1}{2} (3+i n);\frac {1}{2} (5+i n);\frac {1}{2} (1-i a x)\right )}{a (3 i-n) \sqrt {1+a^2 x^2}} \]
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Rubi [A]
time = 0.07, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {5184, 5181, 71}
\begin {gather*} -\frac {2^{\frac {3}{2}-\frac {i n}{2}} \sqrt {a^2 c x^2+c} (1-i a x)^{\frac {1}{2} (3+i n)} \, _2F_1\left (\frac {1}{2} (i n-1),\frac {1}{2} (i n+3);\frac {1}{2} (i n+5);\frac {1}{2} (1-i a x)\right )}{a (-n+3 i) \sqrt {a^2 x^2+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 71
Rule 5181
Rule 5184
Rubi steps
\begin {align*} \int e^{n \tan ^{-1}(a x)} \sqrt {c+a^2 c x^2} \, dx &=\frac {\sqrt {c+a^2 c x^2} \int e^{n \tan ^{-1}(a x)} \sqrt {1+a^2 x^2} \, dx}{\sqrt {1+a^2 x^2}}\\ &=\frac {\sqrt {c+a^2 c x^2} \int (1-i a x)^{\frac {1}{2}+\frac {i n}{2}} (1+i a x)^{\frac {1}{2}-\frac {i n}{2}} \, dx}{\sqrt {1+a^2 x^2}}\\ &=-\frac {2^{\frac {3}{2}-\frac {i n}{2}} (1-i a x)^{\frac {1}{2} (3+i n)} \sqrt {c+a^2 c x^2} \, _2F_1\left (\frac {1}{2} (-1+i n),\frac {1}{2} (3+i n);\frac {1}{2} (5+i n);\frac {1}{2} (1-i a x)\right )}{a (3 i-n) \sqrt {1+a^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 117, normalized size = 0.98 \begin {gather*} \frac {2^{\frac {3}{2}-\frac {i n}{2}} (1-i a x)^{\frac {3}{2}+\frac {i n}{2}} \sqrt {c+a^2 c x^2} \, _2F_1\left (\frac {1}{2} (3+i n),\frac {1}{2} i (i+n);\frac {1}{2} (5+i n);\frac {1}{2} (1-i a x)\right )}{a (-3 i+n) \sqrt {1+a^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int {\mathrm e}^{n \arctan \left (a x \right )} \sqrt {a^{2} c \,x^{2}+c}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {c \left (a^{2} x^{2} + 1\right )} e^{n \operatorname {atan}{\left (a x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\mathrm {e}}^{n\,\mathrm {atan}\left (a\,x\right )}\,\sqrt {c\,a^2\,x^2+c} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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